At this point the admissibility conditions for the existence of a DSW need to be checked
[1]}, [2]}. These relate to the genuine nonlinearity and hyperbolicity of the associated Whitham
modulation equations upon which DSW solutions are ultimately based. The breakdown of these lead to linear degeneracy in the
first case and modulational instability in the second case, so that the standard DSW solution, as derived above, breaks down.
The admissibility conditions require that the derivatives of the velocities of the trailing edge \(s_{i}\)  and of the
leading edge \(s_{+}\)  do not have turning points as functions of \((u_{-}\) ,\(u_{+})\) . It is found from the leading edge velocity
expression (REF ) for \(s_{+}\)  that these derivatives do not vanish at the leading edge, so there is no breakdown
of the DSW structure at the leading edge. At the trailing edge, the derivative of \(s_{i}\)  with respect to \(u_{-}\)  vanishes when
\(u_{+} = u_{-} -3/32\) , which is \(u_{+} = 0.90625\)  for \(u_{-}=1\) , and the derivative with respect to \(u_{+}\)  vanishes when
\(u_{+} = u_{-} - 5/32\) , which is \(u_{+} = 0.84375\)  for \(u_{-}=1\) . In these calculations, \(C_{4}\)  given by (REF ) has been
approximated by \(C_{4} = 3/2\)  as \(\nu \)  is large. The turning point with respect to \(u_{-}\)  means that after this point a simple
wave solution of the Whitham modulation equations governing the DSW is not possible, while the vanishing of the derivative with
respect to \(u_{+}\)  means that the Whitham modulation equations lose hyperbolicity, so that the modulated wavetrain is not stable.
The breakdown of these admissibility conditions is usually clearly mirrored in numerical solutions by non-standard DSW behaviour,
such as the generation of a multi-phase wavetrain or wavetrain instability [2]}. However, for the nematic DSW there is
no clear evidence of such behaviour for values of \(u_{+}\)  below the turning points, as seen in Figures REF (a) and (b).
These figures show some evidence of instability at the trailing edge of the DSW, but it is minor.
